Question: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{n^2 + 5n - 24}{n^2 + 3n - 18}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{n^2 + 5n - 24}{n^2 + 3n - 18} = \dfrac{(n + 8)(n - 3)}{(n + 6)(n - 3)} $ Notice that the term $(n - 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n - 3)$ gives: $t = \dfrac{n + 8}{n + 6}$ Since we divided by $(n - 3)$, $n \neq 3$. $t = \dfrac{n + 8}{n + 6}; \space n \neq 3$